By the classification of formal groups in characteristic $p$, we know that isomorphism classes of connected smooth $1$-dimensional formal groups, equivalently group scheme structures on $\operatorname{Spec} \bar{\mathbb
F}_p[[x]]$, are in a bijection, defined by the height invariant, with $\mathbb N \cup \infty$ .
I understand the formal groups of height $1$ and $\infty$ quite well – they are the germs of the additive and multiplicative groups. There are extremely simple explicit formulas for the group laws, or rather the group laws of some example.
I understand the formal groups of height $2$ fairly well. They are the germs of supersingular elliptic curves. This gives a procedure to compute the power series, but not a very easy one.
What about formal groups of other heights? Is it possible to give an explicit formula for the coefficients of the power series? How difficult are they to compute? Can the power series be taken to be algebraic functions?
Best Answer
A few years ago I computed some formal group laws over ${\mathbb F}_2$ of heights 2, 3, and 4. I've just put the resulting pictures online:
I find the patterns fascinating because there is a fractal element to them (patterns are repeated at different scales). I also wonder if one can define a limit that captures the large scale look of these pictures.
As for the mathematical question whether & how higher height formal group laws occur in nature, you might like Jan Stienstra's "Formal group laws arising from algebraic varieties". He computes the formal Brauer group of a K3 surface. If I recall this correctly that can give you formal group laws up to height 10.